Control of MISO node

ABSTRACT

Multiple-Input-Single-Output (MISO) devices and associated VPA control algorithms are provided herein. For example, a method includes receiving a plurality of control signals. The method also generates a plurality of substantially constant envelope signals from the plurality of control signals and a reference signal. The method combines the plurality of substantially constant envelope signals at a multiple input single output (MISO) node to generate a desired waveform. Further, the method controls the desired waveform at the MISO node based on a signal constellation corresponding to the plurality of control signals.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 12/628,510, filed Dec. 1, 2009, now allowed, which is acontinuation of U.S. patent application Ser. No. 12/142,521, filed Jun.19, 2008, now U.S. Pat. No. 8,013,675, which claims the benefit of U.S.Provisional Patent Application No. 60/929,239, filed Jun. 19, 2007, andU.S. Provisional Patent Application No. 60/929,584, filed Jul. 3, 2007,which are all incorporated herein by reference in their entireties.

The present application is related to U.S. patent application Ser. No.11/256,172, filed Oct. 24, 2005, now U.S. Pat. No. 7,184,723 and U.S.patent application Ser. No. 11/508,989, filed Aug. 24, 2006, now U.S.Pat. No. 7,355,470, both of which are incorporated herein by referencein their entireties.

BACKGROUND

1. Field

Embodiments of the present invention are related generally to poweramplification, modulation, and transmission.

2. Background

Outphasing amplification techniques have been proposed for RF amplifierdesigns. In several aspects, however, existing outphasing techniques aredeficient in satisfying complex signal amplification requirements,particularly as defined by wireless communication standards. Forexample, existing outphasing techniques employ an isolating and/or acombining element when combining constant envelope constituents of adesired output signal. Indeed, it is commonly the case that a powercombiner is used to combine the constituent signals. This combiningapproach, however, typically results in a degradation of output signalpower due to insertion loss and limited bandwidth, and, correspondingly,a decrease in power efficiency.

What is needed therefore are power amplification methods and systemsthat solve the deficiencies of existing power amplifying techniqueswhile maximizing power efficiency and minimizing non-linear distortion.Further, power amplification methods and systems that can be implementedwithout the limitations of traditional power combining circuitry andtechniques are needed.

BRIEF SUMMARY

Multiple-Input-Single-Output (MISO) amplification and associated VPAcontrol algorithms are provided herein. In an embodiment, a methodincludes receiving a plurality of control signals. The method alsogenerates a plurality of substantially constant envelope signals fromthe plurality of control signals and a reference signal. The methodcombines the plurality of substantially constant envelope signals at amultiple input single output (MISO) node to generate a desired waveform.Further, the method controls the desired waveform at the MISO node basedon a signal constellation corresponding to the plurality of controlsignals.

In another embodiment, an apparatus includes one or more modulators, amultiple input single output (MISO) device, and a control circuit. Theone or more modulators are configured to receive a plurality of controlsignals and to generate a plurality of substantially constant envelopesignals from the plurality of control signals and a reference signal.The MISO device is configured to combine the plurality of substantiallyconstant envelope signals at a MISO node to generate a desired waveform.Further, the control circuit is configured to control the desiredwaveform at the MISO node based on a signal constellation correspondingto the plurality of control signals.

Further embodiments, features, and advantages of the present invention,as well as the structure and operation of the various embodiments of thepresent invention, are described in detail below with reference to theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES

Embodiments of the present invention will be described with reference tothe accompanying drawings, wherein generally like reference numbersindicate identical or functionally similar elements. Also, generally,the leftmost digit(s) of the reference numbers identify the drawings inwhich the associated elements are first introduced.

FIG. 1 illustrates two example lossless combiner topologies.

FIG. 2 illustrates an example MISO topology.

FIG. 3 illustrates a conventional branch amplifier topology with alossless combiner.

FIG. 4 illustrates an example MISO amplifier topology.

FIG. 5 illustrates a traditional topology with lossless combiner.

FIG. 6 illustrates an example MISO topology.

FIG. 7 illustrates an S-parameter test bench for a lossless T-line(lossless T).

FIG. 8 illustrates an example plot of branch impedance for a lossless T.

FIG. 9 illustrates an example plot of branch phase shift for a losslessT.

FIG. 10 illustrates an S-Parameter Smith Chart for a lossless T.

FIG. 11 illustrates an example plot of the phase shift between combinerinputs for a lossless T.

FIG. 12 illustrates an example plot of gain and phase shift versusfrequency for a lossless T.

FIG. 13 illustrates an example S-parameter test bench for a MISOcombiner node.

FIG. 14 illustrates an example S-parameter Smith Chart for a MISOcombiner node.

FIG. 15 illustrates an example plot of phase shift for a MISO combinernode.

FIG. 16 illustrates a sequence of example Smith Chart plots for alossless Wilkinson T combiner.

FIG. 17 illustrates a sequence of example Smith Chart plots for apseudo-MISO combiner node.

FIG. 18 illustrates an example S-parameter test bench for a MISOamplifier using ideal switching elements.

FIG. 19 illustrates a theoretical perfect outphasing transfercharacteristic along with a simulated transfer characteristic.

FIG. 20 illustrates the behavior of an example MISO amplifier from anefficiency point of view for a variety of control techniques.

FIG. 21 illustrates an example ADS test bench for an example MISOamplifier with blended control.

FIG. 22 illustrates an example ADS test bench for a modified Chireixcombiner.

FIG. 23 illustrates the performance of two example Chireix designs andtwo example MISO designs.

FIG. 24 illustrates example MISO amplifier efficiency versus outputpower for various control and biasing techniques.

FIG. 25 illustrates control bias points in an example VPA using anexample MISO amplifier.

FIG. 26 illustrates an example efficiency performance plot of the systemshown in FIG. 25 for various control techniques.

FIG. 27 illustrates example MISO control and compensation for a rampeddual sideband-suppressed carrier (DSB-SC) waveform.

FIGS. 28 and 29 illustrate actual waveforms from a VPA with an exampleMISO amplifier of a ramped DSB-SC signal along with the MISO/Driver biascontrol signal and the MISO/Driver collector current.

FIG. 30 illustrates an example WCDMA signal constellation.

FIG. 31 illustrates an example “starburst” characterization andcalibration constellation.

FIG. 32 illustrates a single starburst spoke of the example starburstconstellation of FIG. 31.

FIG. 33 illustrates example error surfaces.

FIG. 34 illustrates the relationship between error compensation andcontrol functions for the example of FIG. 27.

FIG. 35 illustrates the relationship between the upper and lower branchcontrol, phase control, and vector reconstruction of signals in complexplane.

FIG. 36 illustrates the interrelationship between various example VPAalgorithms and controls.

FIG. 37 illustrates an example RL circuit.

FIG. 38 illustrates the relationship between the current through theinductor and the voltage across the inductor in the example RL circuitof FIG. 37 for a 9.225 MHz carrier rate.

FIG. 39 illustrates the relationship between the current through theinductor and the energy stored in the inductor in the example RL circuitof FIG. 37 for a 9.225 MHz carrier rate.

FIG. 40 illustrates the relationship between the current through theinductor and the voltage across the inductor in the example RL circuitof FIG. 37 for a 1.845 GHz MHz carrier rate.

FIG. 41 illustrates the relationship between the current through theinductor and the energy stored in the inductor in the example RL circuitof FIG. 37 for a 1.845 GHz carrier rate.

FIG. 42 illustrates another example RL circuit.

FIG. 43 illustrates the currents through the switch branches in theexample RL circuit of FIG. 42 for 90 degrees of outphasing.

FIG. 44 illustrates the current through the inductor in the example RLcircuit of FIG. 42 for 90 degrees of outphasing.

FIG. 45 illustrates the current through the switch branches in theexample RL circuit of FIG. 42 for 180 degrees of outphasing.

FIG. 46 illustrates the current through the inductor in the example RLcircuit of FIG. 42 for 180 degrees of outphasing.

FIG. 47 illustrates the increase in the ramp amplitude of the inductorcurrent for two outphasing angles in the example circuit of FIG. 42.

FIG. 48 illustrates the reactive power cycled in the inductor versus theoutphasing angle in the example circuit of FIG. 42.

FIG. 49 illustrates an example RL circuit.

FIG. 50 illustrates current through the inductor of the example RLcircuit of FIG. 49 for various outphasing angles.

FIG. 51 illustrates current pulses through the inductor of the exampleRL circuit of FIG. 49 for various outphasing angles with switchresistance varied.

FIG. 52 illustrates an example circuit having a load coupled by acapacitor.

FIG. 53 illustrates the currents through the inductor and the load inthe example circuit of FIG. 52.

FIG. 54 illustrates the effect of varying the switch source resistanceson the currents through the inductor and the load in the example circuitof FIG. 52.

FIG. 55 illustrates an example plot of power output versus outphasingangle for a blended outphasing approach.

FIG. 56 illustrates a histogram associated with a WCDMA waveform.

FIG. 57 illustrates the power output to the load in the example circuitof FIG. 52.

FIG. 58 illustrates the average DC current from battery in the examplecircuit of FIG. 52

DETAILED DESCRIPTION

1. Overview

2. Combiner Definition

3. Outphasing and “Lossy” versus “Lossless” Combiner Attributes

4. Output Branch Amplifiers and MISO Amplifier

5. Simplified Laplace Transfer Functions

6. “Lossless” T and MISO Simulation

7. MISO Amplifier Simulations and “Lossless” T Efficiency

-   -   (a) Switching MISO design        -   (i) Circuit Performance            8. Lab Results for MISO Amplifiers and VPA System            9. VPA Waveform Processing with Blended Control Functions            and Compensation    -   (a) Calibration and Compensation        10. Comments on Transient Solutions and a Mathematical Basis for        the MISO Node Operation    -   (a) R-L MISO Circuit without Load and Variable Duty Cycle    -   (b) Blended Outphasing in the R-L Case Without Load    -   (c) Equations Related to Outphasing with a Load        -   (i) Blended Outphasing Efficiency            11. Summary            1. Overview

A comparison of MISO (multiple input single output) amplifierarchitectures employing an innovative Vector Power Amplification (VPA)Technology versus conventional outphasing approaches, such as LinearAmplification using Nonlinear Components (LINC), is provided herein.Description of MISO and VPA architectures are presented in U.S. Pat.Nos. 7,184,723 and 7,355,470, both of which are incorporated byreference in their entireties. This comparison is based in simulationresults as well as theory. Some concepts are especially expanded on forpurposes of this comparison. In particular, since the combining node isa focus of this genre of power amplifier (PA) and is key for manyperformance metric comparisons, its operation is illustrated in adetailed manner. In addition, an important theme of the comparison isefficiency and its relationship to the combining node operation as wellas certain MISO principles.

In sections 2-6, a comparison between MISO) and “Lossless” combiners isprovided. Section 7 provides simulations which illustrate the efficiencyof “Lossless” T and MISO for various blended control MISO approaches.Section 8 provides actual experimental verification of MISO operationand performance, in support of the results presented in Section 7.Section 9 provides the concept of a unified system approach foroptimizing the deployment of MISO. Section 10 provides a mathematicalview of MISO operation. A summary is presented in Section 11.

2. Combiner Definition

Generally, a combiner is a structure which permits the transfer ofenergy from two or more branches or inputs to an output. In contrast,MISO includes structure which facilitates the efficient transfer ofenergy from a power source to a load. Note that the power source maypossess any spectral content, and that this does not require that MISOtransfer power at all from its inputs to its output. Indeed, MISO inputscan be viewed as control points that create a variable impedance at thecombining node or that permit steering currents to enter or leave thenode. As such, in the case of ideal switches at the input branches, nopower is transferred through the input branches. This is in directcontrast to all conventional outphasing systems which depend onefficient transfer of power through the input branches of the combiner.

There are two categories of combiners generally, “lossless” and “lossy.”The industry literature distinguishes between the two categories bynoting that lossy combiners provide isolation between the input portsand that lossless combiners do not. In this disclosure, a losslesscombiner may or may not possess some form of isolation. In particular,significant time delay is considered a certain type of isolation atcertain frequencies.

MISO may be described as lossless. However, as the term “lossless” asused by the industry is often misunderstood or misappropriated, it isuseful to provide some insight into conventional combiner propertiescompared to MISO.

First, it should be noted that lossiness refers to the insertion lossproperties of the combiner structure, which combines the branches of anoutphasing amplifier to generate the amplifier's output. It is alsonoted that loss is a function of frequency for most circuits involvingnon-zero branch impedances. While it may be true that a losslesstransmission line (lossless T) has little insertion loss at DC andcertain harmonically related frequencies associated with thetransmission line branches, it is not lossless at all frequencies forfixed termination impedances. As such, lossless Ts typically possess anoperational bandwidth.

Two versions of the classical “lossless” combiner often reported inliterature are illustrated in FIG. 1. In particular, the top diagramillustrates a “lossless” Wilkinson combiner and the bottom diagramillustrates a Chireix combiner. In contrast, FIG. 2 illustrates a2-input MISO topology with load.

Some fundamental differences can be noted between the MISO topology andthe classical “lossless” combiner topologies. In particular, differencesthat relate to impedance, time delay, and frequency response can benoted. Other differences will be apparent to persons skilled in the art.

From one aspect, the MISO topology possesses branch impedances that aresubstantially equal to zero and thus do not alter phase or amplitudecharacteristics of the branch currents. This results in no restrictionin terms of the frequency response of the topology. In contrast, the“lossless” combiner topologies are in fact lossless at only the designfrequency and certain multiples thereof, and thus causefrequency-dependent amplitude as well as phase modifications on thebranch currents. In fact, significant phase shifts can be experienced atoffsets from the fundamental carrier, which considerably restricts theusable frequency response of the topologies.

From another aspect, the “lossless” combiner topologies utilizereflected wave fronts of transverse-electromagnetic energy (TEM) storedin the transmission lines to constructively or destructively interferein a precise manner, based on wavelength, to provide time delayisolation between the inputs. In contrast, the MISO topology does notstore energy temporally and treats all wavelengths equivalently.

In view of the above, “lossless” combiner topologies have limitedbandwidths. While it is possible to design an array of wideband“lossless” combiners, “lossless” combiners can never possess all of thefollowing attributes: small size, low complexity, low cost, and can beimplemented in IC technology. However, as will be further shown below,the MISO topology provides a substantial improvement, or is evenoptimal, with respect to any one of these attributes compared to“lossless” combiner topologies.

3. Outphasing and “Lossy” versus “Lossless” Combiner Attributes

A variety of amplifier architectures have been analyzed in theliterature, such as LINC, to evaluate the tradeoffs between lossless andlossy combiners. In particular, Conradi, Johnston, and McRory,“Evaluation of a Lossless Combiner in a LINC Transmitter,” Proceedingsof the 1999 IEEE Canadian Conference on Electrical and ComputerEngineering, May 1999, found that a lossy matched combiner (whichprovides some branch isolation) provides better baseline linearityperformance in the overall LINC output response of a LINC architecturethan a lossless combiner. However, the evaluation found that thelossless combiner provides better efficiency. This can be explained bythe fact that a lossy combiner (e.g., Wilkinson) includes a resistor inits topology, which results in power being dissipated in the combiner,adversely affecting the efficiency of the combiner but providing inputbranch isolation.

MISO optimizes efficiency. In an embodiment, linearity is achieved bymeans such as VPA Technology used in conjunction with a MISO amplifier.

In terms of efficiency, lossy combiner architectures are not competitivein today's cellular and mobile wireless market. While other techniquesmay be employed with lossy combiner architectures to trim efficiencylosses in the architecture, these techniques often introducenon-linearities.

In embodiments, MISO designs opt for high efficiency combinerarchitecture and combine characterization, compensation, and/orcalibration to address time variant system non-linearities, which aretypically associated with transistor technology. As such, MISO enhancesefficiency while dedicating digital and/or analog algorithms tolinearize the entire VPA architecture, not just the combiner. Further,the necessary combiner properties are maintained without isolation byemploying VPA Technology to adjust the combiner sources that drive thecombiner's inputs.

4. Output Branch Amplifiers and MISO Amplifier

In traditional architectures, the branch amplifiers are treated asindividual entities with unique inputs and outputs that can be describedin terms of their respective transfer characteristics. Whenever theseclassical amplifiers are employed with a lossy combiner, analysis ismanageable provided that the branch amplifiers are at constant bias.However, if conventional amplifiers are employed with a losslesscombiner, then the analysis becomes more complicated at the circuitlevel if complete transistor models are used. That is why system levelanalysis and characterization are often preferred.

A MISO amplifier is intricate at the circuit level as well but simplerto analyze than a conventional amplifier using a lossless combiner. Infact, embodiments of the MISO amplifier can be viewed as a single unitwith multiple inputs and a single output, with the inputs interacting ornot. The single output of the exemplary MISO amplifier is a complicatedfunction of the inputs, at the detailed circuit level. However, systemlevel characterization of the MISO amplifier can be simplified when thecombiner is viewed as a variable impedance node, with impedance given asa function of outphasing angle or other MISO state. This variableimpedance node collects the sum of multiple branch steering currentswhenever a power source is available at the summing node. The combinernode uses several means to efficiently integrate the branch currentsinto a single branch. For example, outphasing, bias control, powercontrol, amplitude control, and/or impedance control can be used. In anembodiment, in the limiting case, when the MISO amplifier input branchesare modulated as nearly ideal switches, the combiner node inputwaveforms are symmetrically pulse width modulated as a function of theoutphasing angle. In an embodiment, the modulated pulses repeat at thecarrier rate or at some integrally related submultiples or multiple.Further, the pulses are harmonically rich. Therefore, a frequencyselective network is utilized to isolate and match the fundamental ordesired harmonic to the load.

According to an embodiment of the present invention, a MISO amplifierintegrates all of the features of a switching amplifier with the summingnode in a single efficient structure. A traditional LINC amplifier, onthe other hand, treats the amplifiers in each branch as separateentities to be optimized prior to the cascade of the power combiner.

Raab et al., “RF and Microwave Power Amplifier and TransmitterTechnologies—Part 3,” High Frequency Electronics 2003, provides anoverview of classes of amplifiers which can be used with the LINCarchitecture. Similarly, MISO architecture embodiments can use all ofthese classes of operation. However, in an embodiment, the highestefficiency is attained whenever the branch amplifiers and the MISOamplifier are operated in switch mode.

Consider the two circuit topologies in FIGS. 3 and 4. FIG. 3 illustratesa conventional branch amplifier topology with a lossless combiner. FIG.4 illustrates a MISO amplifier topology according to an embodiment ofthe invention. An important difference between the two topologiesrelates to isolation between the branches. It is noted, for example,that currents at the combining node interact through bandwidth dependentcomplex impedances in the lossless combiner topology. In contrast, theMISO amplifier topology does not suffer from this restriction. Thisdifference becomes even more important when switch mode operation isinvoked. Indeed, the currents flowing from the combiner node through theMISO amplifier can be virtually instantaneously interactive with theirbranch switching elements. Traditional topologies, on the other hand, donot permit this instantaneous interaction, with either the switchelements or the load branch. In this sense, the lossless T possesses aform of isolation, known as time isolation, while the MISO amplifiergenerates steering currents or summing currents which contend for thenode without isolation.

Switching transient terms and harmonic content are major considerationsfor the efficiency of a switching amplifier. If the transients arelimited or if the harmonic content is misused, it is not possible forthe amplifier to operate efficiently. Indeed, the literature is repletewith analyses and examples for Class “E” and Class “F” amplifiers whichrequire from 3 to 10 harmonics along with proper management of theconduction angles for current and voltage to obtain a desired result. Itis important therefore to note that harmonics are not altered by thecombiner transfer function in a MISO amplifier, according to anembodiment of the present invention.

5. Simplified Laplace Transfer Functions

The combiner topologies of FIGS. 3 and 4 can be further analyzed usingthe circuits of FIGS. 5 and 6. FIG. 5 illustrates a traditional topologywith lossless combiner. FIG. 6 illustrates an example MISO topologyaccording to an embodiment of the present invention.

Analyzing the circuit of FIG. 5 in the Laplace domain yields:

V 0 ⁡ ( s ) V S ⁡ ( s ) = ( 1 + ⅇ - T M ⁢ s ) ⁢ ( ( C + S ) ⁢ R L ( C + S ) ⁢( ( M + R L ) + ( M + R L + C + S ) ) )

V_(SU)(s)=V_(SL)(s)e^(−T) ^(M) ^(s) (Upper and Lower Branch SourceTransform)

V_(S) ΔV_(SU)

T_(M) Δ Delay difference between Upper and Lower Branch Input Signalsrelated to Modulation and Outphasing Angle.

T_(C) Δ Carrier Cycle Time

_(C) Δ Combiner Branch Impedance (T-line description for “Lossless”Wilkinson or other “Lossless” T.)

_(S) Δ Impedance of Upper or Lower Branch Source assumed to be the same.

In these models, the upper and lower branch sources are identical exceptfor the time delay (phase term) related by T_(M). T_(M) can varyaccording to the modulation envelope. It is possible to createmodulation angles such that T_(M) forces a transfer function whichapproaches zero. An alternate form of the simplified Laplace TransferFunction of the topology of FIG. 5 can be written as:

V 0 ⁡ ( s ) V S ⁡ ( s ) = ⅇ - τ D ⁢ s ⁡ ( 1 + ⅇ - T M ⁢ s ) ⁢ ( ( C ′ + S ) ⁢ RL ( C ′ + S ) ⁢ ( ( M + R L ) + ( M + R L + C ′ + S ) ) )

′_(C) Δ Impedance Transformation for the “Lossless” Combiner Bench

τ_(D) Δ Effective Delay through the “Lossless” Combiner Branch

In this form it can be seen that the exponent, τ_(D), for combinerbranch delay forces different impedance values at different harmonics ofthe carrier, which in turn modifies V₀(s)/V_(S)(s) at those harmonics.

The transfer function of the MISO topology of FIG. 6 can be written as:

R L ⁢ i L ⁡ ( s ) V S = R L s ⁡ ( α 1 ⁡ ( s ) ⁢ ( pu + R a U a ⁡ ( s ) ) - α 0⁡( s ) ⁢ R b U a ⁡ ( s ) ) α 1 ⁢ ⁢ ( s ) = ( U a ⁡ ( s ) R a + U b ⁡ ( s ) R b) ⁢ ( R b U b ⁡ ( s ) + R L + M ) - R b ⁢ U a ⁡ ( s ) R a ⁢ U b ⁡ ( s ) α 0 ⁡ (s ) = U b ⁡ ( s ) R b ⁢ ( R b U b ⁡ ( s ) + R L + M )$R_{{SX}_{a}} = \frac{R_{a}}{U_{a}(s)}$$R_{{SX}_{b}} = \frac{R_{b}}{U_{b}(s)}$${U_{a}(s)} = {\sum\limits_{k = 0}^{\infty}{( {{\mathbb{e}}^{- {ks}} - {\mathbb{e}}^{{- {s{({k + {1/2}})}}}T_{C}}} )\frac{1}{s}}}$${U_{b}(s)} = {\sum\limits_{k = 0}^{\infty}{( {{\mathbb{e}}^{- {ks}} - {\mathbb{e}}^{{- {s{({k + {1/2}})}}}T_{C}}} )\frac{{\mathbb{e}}^{- {s{(T_{M})}}}}{s}}}$

In the MISO model, a main difference is the fact that rather thanvoltage sources which combine power via the combiner branches, the powertransfer occurs from the battery to the combiner node by the operationof switched resistances on the combiner inputs. These switching cyclesgenerate currents, causing energy storage in

_(pu). The energy is then released to the load, when the switches open.The combiner inputs do not transfer power whenever R_(a)=R_(b)=0,∞.

Although V₀(s)/V_(S)(s) was calculated above, for convenience, it shouldbe apparent based on the teachings herein that I₀(s)/I_(S)(s) can beobtained from the relationships:

${I_{0}(s)} = \frac{V_{0}(s)}{R_{L}}$ I S ⁡ ( s ) = V S ⁡ ( s ) S ⁢ ( s )

T_(M) varies according to the outphasing angle and correspondinglyforces a different impedance at the combiner node. This is furtherexamined in section 6 below, which illustrates this phenomenon withS-parameters.

6. “Lossless” T and MISO Simulation

A simulation using ADS (Advanced Design System) reveals the differencesbetween conventional lossless combiner topologies and example MISOtopologies from an S-Parameter (Scattering parameter) point of view.

FIG. 7 illustrates an S-parameter test bench for a lossless T, whichuses a common compound transmission line. Simulations results using theS-parameter test bench of FIG. 7 are illustrated in FIGS. 8-12.

FIG. 8 illustrates an example plot of branch impedance for a lossless T.In particular, FIG. 8 illustrates the branch impedance associated withS31 and shows that the branch impedance associated with S31 varieswildly over the frequency range of several harmonics. Three (3)harmonics are marked for reference. FIG. 9 illustrates an example plotof branch phase shift for a lossless T. In particular, FIG. 9illustrates the branch phase shift associated with S31 and shows thesignificant phase shift at each harmonic. It is noted that the phaseshift is not modulo λ and that S31 traces multiple loops around theSmith Chart for 3 harmonics.

FIG. 10 illustrates an S-Parameter Smith Chart for a lossless T. Inparticular, FIG. 10 illustrates a S-Parameter Smith Chart for S(1, 2),S(1, 1), S(1, 3). FIG. 11 illustrates the phase shift associated withS12, which is the phase shift between the combiner inputs. It isimportant to note the multiple loop trace for S12 to accommodate 3harmonics. Also to be noted are the drastic variations in both gain andphase shift associated with S33, as illustrated in FIG. 12.

FIG. 13 illustrates an S-parameter test bench for a MISO combiner node.Simulation results using the S-parameter test bench of FIG. 13 areillustrated in FIGS. 14-15.

FIG. 14 illustrates an example S-parameter Smith Chart for a MISOcombiner node. In particular, FIG. 14 illustrates an S-parameter SmithChart for S12, S13, and S11 (S13=S12). FIG. 15 illustrates an exampleplot of phase shift for a MISO combiner node. In particular, FIG. 15illustrates the phase shift associated with S31. As shown, the phaseshift is minimal and the impedance is constant over the entire range offrequencies. This also holds true for the combiner path to the outputload and for the combiner inputs as well.

Also, from this analysis, note that S12=S21=S31 for the MISO combinernode and that there is no phase shift or time delay in the branches.

FIGS. 16 and 17 illustrate sequences of Smith Chart plots, which showhow the insertion S parameters change as a function of outphasing anglesfor three harmonics. In particular, FIG. 16 illustrates the case of alossless Wilkinson T combiner. FIG. 17 illustrates the case of apseudo-MISO combiner node. As shown in FIG. 16, in the case of thelossless Wilkinson T, the insertion delay is considerable at eachharmonic. It is noted that the sweep spans 90° of outphasing.

Also noted is the difference between the lossless T and the pseudo-MISOfor S12 at the fundamental at 1.84 GHz. Also with respect to theharmonics, the lossless T possesses an alternating origin for the locusof points depending on whether the harmonics are even or odd, inaddition to the other apparent differences from the pseudo-MISO case. Incontrast, in the MISO case, all harmonics launch or spawn from a commonpoint on the real axis. This is significant for asymmetric pulses inswitching devices. Indeed, efficient output PA device designs willpossess some level of pulse asymmetry to run in an efficient manner.This asymmetry must be accommodated for practical applicationparticularly as outphasing is applied. This asymmetry is more difficultto manage in the lossless T case, for some switching amplifiertopologies.

7. MISO Amplifier Simulations and “Lossless” T Efficiency

According to an embodiment, the VPA Architecture utilizes a MISOamplifier to enable efficient operations while maintaining a myriad ofother benefits. The manner in which the MISO is implemented oftenutilizes a combiner node with little or no substantial isolation betweenthe inputs and the driving amplifiers or sources. Nevertheless,appropriate current summing at the node is accomplished by controllingthe properties of the source amplifiers through a blend of outphasing,bias control, amplitude control, power supply control, and impedancecontrol. In this manner, suitable transfer functions can be constructedto enable efficient transmitter operation, even with waveforms whichpossess large PAPR (Peak to Average Power Ratio).

One method of accomplishing the desired MISO operation is by blendingbias control along with outphasing. Another method could be a blend ofamplitude control as well as outphasing and bias control. It is alsopossible to combine power supply modulation if so desired. The followingsections illustrate MISO operation via simulation in two basicconfigurations. The first set of simulation results relate to aswitching MISO architecture captured in the ADS tool using “ideal”switches. The second set of simulation results provides a comparisonbetween the MISO and more traditional lossless combiners using quasiClass “C” amplifier cores. The focus of the comparisons is efficiencyversus power output for various levels of output back off. The basicimplementation of back off comes from outphasing for the lossless Tcombiner examples while the MISO examples use additional degrees offreedom to accomplish a similar function.

As will be further described below, the simulations illustrate that theMISO node enables efficient multi branch amplifier signal processingwhen combined with VPA Technology.

(a) Switching MISO Design

A simulation was conducted in ADS to illustrate the operation of a MISOamplifier using “ideal” switching elements. FIG. 18 illustrates thetopology in ADS.

It is noted that the switches represent switching amplifier elementswith virtually infinite gain. Parasitics are zero except for theinsertion resistance of the switch, which can be controlled from aminimum of 0.1Ω to a very large value. The minimum and maximum values ofresistance and controlling function are fully programmable. Thefunctions are deliberately formulated with dependency on outphasingangle, which is swept for many of the simulations using the variableT_(M), which is the time delay between the two source square waveleading edges.

The sources have rise and fall times which can be defined, as well asfrequency amplitude and offset. Simulations were conducted in bothtransient and harmonic balance test benches.

FIG. 19 illustrates a perfect outphasing transfer characteristic alongwith a simulated transfer characteristic. The perfect characteristicrepresents the result:Power Out=10 log₁₀(A cos(φ/2))²+30 dBmwhere φ/2 is ½ the outphasing angle. φ is presented on the horizontalaxis. The blue trace is the ideal result or template while the red traceis the simulated circuit response. The ideal trace is normalized to themaximum power out for the simulated circuit example. Note that theequivalent outphasing response of the combiner is very close to theideal response without detailed calibration. in embodiments, this isaccomplished by providing simple blended control functions and accuracyis typically forced by detailed compensation or calibration.

FIG. 20 illustrates the behavior of an example MISO amplifier from anefficiency point of view for a variety of control techniques. It isnoted that efficiency is poor with pure outphasing control. This isbecause the branch driving impedances for the inputs to the combiner arenot optimized for efficiency versus P_(out). Indeed, the optimalimpedance function is non-linear and time variant and is thereforeaccounted for in the VPA transfer function which is connected to theMISO amplifier.

The other traces in FIG. 20 illustrate several other controls, including“outphase plus bias” control. In practice, this control blend providesvery accurate control for precise reconstruction of waveforms such asEDGE. That is, very good ACPR performance is attained using thistechnique. In addition, it is usually the case that higher output poweris available by employing this type of control.

Another control blend uses power supply control in conjunction withphase and bias.

(i) Circuit Performance

This section provides the results of simulations performed on circuits.The circuits are based on Class “C” amplifiers driven by nearly idealsources, which can be outphased. ADS was utilized as the simulationtool. FIG. 21 illustrates the ADS test bench for a MISO amplifier withblended control. FIG. 22 illustrates the ADS test bench for a modifiedChireix combiner.

FIG. 23 illustrates the performance of two Chireix designs and two MISOdesigns based on the test benches of FIGS. 21 and 22. The cyan tracecorresponds to a MISO amplifier with pure outphasing control. Thecompensated Chireix and classical Chireix are illustrated in green andblue respectively and provide good efficiency versus output powercontrol. Finally, a blended MISO design response is illustrated inviolet, with excellent efficiency over the entire output power controlrange.

Although not illustrated in these simulations, it is also possible toaccentuate the efficiency over a particular range of output power tooptimize efficiency for specific waveform statistics. This feature is afully programmable modification of the basic transfer function andcontrol characteristic equation according to embodiments of the presentinvention.

8. Lab Results for MISO Amplifiers and VPA System

This section provides results of laboratory measurements andexperiments. Results are included for the MISO amplifier as well as theentire VPA 2 branch RF signal processing chain. The results illustratethe same trend of performance disclosed in the previous sectionsdirected to simulation results.

FIG. 24 illustrates MISO amplifier efficiency versus output power forvarious control and biasing techniques, for a CW signal.

In addition to MISO amplifier control techniques, blended controltechniques can also be used for other stages of the VPA. Accordingly, itis also possible to control bias points along the upper and lower branchamplifier chain, as illustrated in FIG. 25, for example. This allows forthe optimization of the entire VPA system, not just the MISO stage. FIG.26 illustrates the efficiency performance of the system shown in FIG. 25for a variety of control techniques. As shown, these techniques includenot only MISO control techniques but also control techniques applied atother control points of the VPA system.

Typically, blended techniques using outphasing plus bias control providevery good ACPR performance for complicated waveforms, especially overextended dynamic range. In addition, the blended technique generallypermits slightly greater compliant maximum output powers for complicatedlarge PAPR waveforms.

9. VPA Waveform Processing with Blended Control Functions andCompensation

FIG. 25 illustrates a 2 branch implementation of the VPA architecturewith a MISO amplifier. As explained in previous sections, the MISOcombining node utilizes a combination of outphasing, bias control,amplitude control, power supply control, and impedance control toaccomplish an efficient and accurate transfer characteristic. The blendthat is selected for the control algorithm is a function of controlsensitivity, waveform performance requirements, efficiency, hardware,firmware complexity, and required operational dynamic range. Accordingto embodiments of the present invention, EDGE, GSM, CDMA2000, WCDMA,OFDM, WLAN, and several other waveforms have been demonstrated on aunified platform over full dynamic range in a monolithic SiGeimplementation, using bias and outphasing control.

This section explains some aspects of the bias and amplitude control andprovides the basis for understanding compensation associated with theVPA. This system level perspective is related to the MISO operation.Since the MISO amplifier consumes a substantial portion of power of allfunction blocks in the VPA system it must be operated and controlledcorrectly in the context of efficient waveform reconstruction. Thisdemands a symbiotic relationship between the MISO amplifier, drivers,pre-drivers, vector modulators, and control algorithms.

FIG. 27 illustrates MISO control and compensation for a ramped dualsideband-suppressed carrier (DSB-SC) waveform, according to anembodiment of the present invention. The plots shown in FIG. 27 aregenerated using actual lab results.

The top plot in FIG. 27 illustrates the envelope for desired a linearlyramped RF waveform. The envelope represents the magnitude of thebaseband waveform prior to algorithmic decomposition. Although notexplicitly indicated in this set of plots the RF waveform changes phase180° at the zero crossing point (sample 200). The three subsequent plotsin FIG. 27 show the control functions required to reproduce the desiredresponse at the filtered MISO output. In particular, the second plot isa plot of outphasing angle. The upper trace is a reference plot of theoutphasing angle in degrees for a pure “ideal”, outphasing system. Thelower trace illustrates an actual outphasing control portion of ablended outphasing plus bias control algorithm. Note that the fulloutphasing angle is approximately 75°, as opposed to 180° for purelyoutphased systems.

The third plot in FIG. 27 illustrates bias control, which operates inconcert with outphasing control to produce the blended attenuation ofthe ramp envelope all the way to the MISO output. This control includescompensation or calibration for amplitude mismatches, phase asymmetries,etc. Therefore, the control waveforms provided must take into accountthe non-ideal behavior of all of the components and functions in thedual branch chain. Both driver and MISO stages are controlled for thisexample. Controlling the bias of the driver reduces the input drive tothe MISO as a function of signal envelope

The fourth lot at the bottom of FIG. 27 illustrates the required phasecontrol to guarantee phase linearity performance of the waveform overits entire dynamic range. At sample 200 a distinct discontinuity isapparent, which is associated with the 180° inversion at the ramp's zerocrossing. Note that the phase compensation is very non-linear near thezero crossing at sample 200. At this point, the amplifiers are biasednear cut off and S21 is very ill behaved for both the driver and PA/MISOstage. The plot associated with phase compensation represents theadditional control required for proper carrier phase shift given thedrastic shift in phase due to changing bias conditions. It should befurther stated that the relationship between amplitude control and phaseis such that they are codependent. As outphasing and bias are adjustedto create the signal envelope the phase compensation requirements changedynamically.

FIGS. 28 and 29 illustrate actual waveforms from a VPA with MISO,according to an embodiment of the present invention, which show theramped DSB-SC signal along with the MISO/Driver bias control signal andthe MISO/Driver collector current.

(a) Calibration and Compensation

Any control algorithm must contemplate the non ideal behavior of theamplifier chain. AM-PM and PM-AM distortion must also be addressed. Manysignals of interest which are processed by the VPA architecture can berepresented by constellations and constellation trajectories within thecomplex signaling plane. The RF carrier at the output of the filteredMISO can be written in complex form as:y(t)=A(t)e ^(j(ω) ^(C) ^(t+Θ))

ω_(C) Δ Carrier Frequency

ΘΔ Modulation Angle

A(t)Δ Amplitude Modulation

The complex signal plane representations of interest are obtained fromthe Re{y(t)} and Im{y(t)} components. A goal for the VPA is to producesignals at the filtered MISO node such that when decomposed into theircomplex envelope components, and projected onto the complex signalingplane, the distortions are minimal and efficiency is maximized. Anexample for such a signal constellation is given in FIG. 30.

Therefore, if the VPA response is known for the entire complex plane, asuitable algorithm will adjust the control signals to contemplate alldistortions. If the control is accurate enough given the requiredcompensations, compliant signal reconstruction is possible at thecombining node.

Suppose that the VPA system is stimulated in such a way that the outputRF signal can be processed to detect the system non-linearities,superimposed on the complex signaling plane. FIG. 31 illustrates anexample “starburst” characterization and calibration constellation,which is one technique to characterize the non-linearities imposed bythe system as measured at the combining node, after filtering.

The red radials represent stimulus signals in the complex plane. Theradials are swept out to the unit circle and back to the origin for aplethora of angles. Each swept radial at the system input spawns acorresponding blue radial at the output. Notice the bifurcation in theblue radials. This arises due to the memory inherent in high power SiGetransistor devices operated over certain portions of their dynamicrange. It can be inferred from the complex plane starburststimulus/response that a significant amount of amplitude and phasenon-linearity is present. This non-linearity can be captured as an error({right arrow over (D)}_(ε) _(R) ) and characterized.

FIG. 32 illustrates a single starburst spoke of the example starburstconstellation of FIG. 31. As shown in FIG. 32, for each point along thetrajectory, there is a magnitude and phase error or distortion which isrelated to error vector, {right arrow over (D)}_(ε) _(R) . If thecomponents of the error vector are disassembled into magnitude and phasefor the entire complex plane, then error surfaces can be constructed asshown in FIG. 33.

The magnitude error |D_(ε) _(R) | must be taken into account by the biasand outphasing control. The phase compensation must take intoconsideration the phase error ∠D_(ε) _(R) . FIG. 34 illustrates therelationship between error compensation and control functions for theexample of FIG. 27.

Another way to examine the role of the controls in signal reconstructionis to associate the controls with their impact on vector operations inthe complex plane. The vectors are outphasing signals generated in theupper and lower branches of the hardware, which drives and includes theMISO amplifier. The outphasing angle is controlled, along with the gainof the driver and gain of the MISO amplifier.

FIG. 35 illustrates the relationship between the upper and lower branchcontrol, phase control, and vector reconstruction of signals in complexplane.

The upper and lower branch signals at the MISO amplifier inputs arecontrolled in terms of bias and outphasing angle φ to create theappropriate amplitude at the MISO combining node. In addition, theamplitudes of the MISO inputs may be controlled by varying the driverbias. |{right arrow over (D)}_(ε) _(R) | must be taken into account whenimplementing the blended control.

The phase of the reconstructed carrier is given by Θ and is controlledby manipulation of vector modulators at each branch input or controllingthe phase of the RF carrier directly at the RF LO synthesizer for thesystem (or both). This phase Θ is associated with the modulation angleof the application signal, y(t).

Although a distinction is drawn between amplitude control and phasecontrol of Θ they are in fact dependent and interact considerably. Theirrespective calculations and compensations are conjoined just as the realand imaginary components of {right arrow over (D)}_(ε) _(R) areinterrelated. This is due to AM-PM and PM-AM distortion in physicalhardware. As bias is reduced or power supplies varied or impedancestweaked, the insertion phase of both branches of a two branch MISOchanges and therefore is a component of ∠{right arrow over (D)}_(ε) _(R). Hence, generating a particular point or trajectory in the complexplane related to the complex signal, at the filtered MISO output,requires the solution of at least a two dimensional parametric equation.The numbers generated for amplitude control of the complex signal arepart of the solution for Θ, the modulation angle, and vice versa.

FIG. 36 illustrates the interrelationship between various example VPAalgorithms and controls.

10. Comments on Transient Solutions and a Mathematical Basis for theMISO Node Operation

Prior sections have shown that a purely outphased MISO is very efficientat maximum power output for a zero degree outphasing angle. Efficiencywas also shown to drop as outphasing is applied. However, when the MISOamplifier is used in conjunction with VPA principles and controlalgorithms, improved efficiency is maintained throughout the power rangeof interest. This section provides a mathematical treatment of thisbehavior, including differential equation analysis which corroboratesthe simulated data presented previously.

Transient analysis usually requires the solution of a system ofequations. In the continuous or quasi-continuous case, a time domainsolution for t=(0⁺→∞) demands the solution of 2^(nd) order differentialequations with time variant coefficients. Such equations are notpractical to solve in closed form. Nevertheless, some piecewisesolutions and formulation of the problem can yield important insights.

(a) R-L MISO Circuit without Load and Variable Duty Cycle

A principle of the switching amplifier requires that the power sourcetransfers energy to some sort of energy storage device and then removesthe energy (or allow it to “siphon” off) at some time later by efficientmeans for use by a load. The energy storage device may be a capacitor,inductor, transmission line, electrically coupled circuit, magneticallycoupled circuit, or hybrids of all of the above. The transmission lineis often overlooked as a means of storing energy. However, the TEM modesdo in fact transport energy and can be efficient, provided loss tangentsare minimal. The following treatment illustrates the use of an inductorto store energy. Consider the example RL circuit illustrated in FIG. 37.

Suppose the switch is closed at t=0. At t→0⁺ the characteristic equationof the system may be used to determine the current flowing in thecircuit and the energy stored in the inductor. The differential equationis:

${\frac{L{\mathbb{d}i_{a}}}{\mathbb{d}t} + {{Rs}_{X_{A}}i_{a}}} = V_{S}$

The solution of this equation requires the consideration of initialconditions:i _(a)(t=0⁻)=0i _(a)(t=0⁺)=0

Therefore,

$\frac{\mathbb{d}i_{a}}{\mathbb{d}t} = \frac{V_{S}}{L}$

The final current at t→∞ must approach Vs/R_(SX) _(A) .

This yields a solution of:

$i_{a} = {\frac{V_{S}}{R_{{SX}_{A}}}( {1 - {\mathbb{e}}^{{- {({R_{{SX}_{A}}/L})}}t}} )}$

FIG. 38 illustrates the relationship between the current through theinductor and the voltage across the inductor in the example RL circuitof FIG. 37 for a 9.225 MHz carrier rate.

FIG. 39 illustrates the relationship between the current through theinductor and the energy stored in the inductor in the example RL circuitof FIG. 37 for a 9.225 MHz carrier rate.

The circuit uses a 1 volt battery, 5 nH inductor and 1Ω resistor (R_(SX)_(A) ).

FIGS. 38 and 39 are from a simulation that permits the inductor currentto reach a maximum limit. This can be accomplished by controlling theduty cycle and frequency of the switch, such that the switching cycletime is virtually infinite compared to the circuit time constant.

If the switch frequency is required to operate at the carrier rate, thenonly a fraction of the energy may be stored in the inductor given thesame circuit values, if the carrier cycle time is considerably shorterthan the time constant (R_(SX) _(A) /L). For instance, consider thecarrier rate of 9.225 MHz in the example illustrated in FIGS. 38 and 39compared to the example illustrated in FIGS. 40 and 41, which uses aswitch rate of 1.845 GHz, with the same circuit values.

FIG. 40 illustrates the relationship between the current through theinductor and the voltage across the inductor in the example RL circuitof FIG. 37 for a 1.845 GHz MHz carrier rate.

FIG. 41 illustrates the relationship between the current through theinductor and the energy stored in the inductor in the example RL circuitof FIG. 37 for a 1.845 GHz carrier rate.

It is noted from FIGS. 40 and 41 that the current in this time domainrange is approximately a ramp. Also, the average current in the inductoris illustrated in FIG. 40. This is an important consideration forefficiency calculations since the average current through the resistorR_(SX) _(A) causes power to be dissipated. In this example, the resistordissipates approximately 1.6 mW.

Suppose that the R-L switched circuit of FIG. 37 is augmented withanother switch. Further suppose that the two switches can be outphased.The circuit is illustrated in FIG. 42.

Assuming that the time delay between pulses controlling the switchesS_(X) _(A) and S_(X) _(B) is given by T_(M), which is related to theoutphasing angle, there are 4 equivalent resistance values which arerelevant over certain segments of time. These represent 4 unique timeconstants given by:

$\tau_{1} = \frac{R_{{SX}_{A}}}{L}$$\tau_{2} = {( \frac{R_{{SX}_{A}} \cdot R_{{SX}_{B}}}{R_{{SX}_{A}} + R_{{SX}_{B}}} )/L}$τ₃ = (R_(SX_(B))/L) τ₄ = ∞

Taking into account an infinite pulse train, the following cyclic unitstep functions are constructed and associated with each time constant:

$U_{1} = {\sum\limits_{k = 0}^{\infty}\lbrack {( {u( {t - {kT}_{C}} )} ) - {u( {t - ( {{kT}_{C} + T_{M}} )} )}} \rbrack}$$U_{2} = {\sum\limits_{k = 0}^{\infty}\lbrack {( {u( {t - {kT}_{C} + T_{M}} )} ) - {u( {t - ( {T_{C}( {k + {1/2}} )} )} )}} \rbrack}$$U_{3} = {\sum\limits_{k = 0}^{\infty}\lbrack ( {{u( {t - {( {k + {1/2}} )T_{C}}} )} - {u( {t - ( {{T_{C}( {k + {1/2}} )} + T_{M}} )} )}} ) \rbrack}$$U_{4} = {\sum\limits_{k = 0}^{\infty}\lbrack ( {u( {t - ( {{T_{C}( {k + {1/2}} )} + T_{M}} ) - {ut} - {T_{C}( {k + 1} )}} )} ) \rbrack}$$T_{C}{\underset{\_}{\Delta}( {{Carrier}\mspace{14mu}{Rate}} )}^{- 1}$

T_(M) Δ Time Delay associated with outphasing modulation angle.

Inspection reveals that u_(v) do not overlap in time. Therefore thedifferential equation solution will take on the following form fori_(S):

$i_{S} = {\frac{Vs}{L{\sum\limits_{i = 1}^{4}{u_{i}\tau_{i}}}}( {1 - {\mathbb{e}}^{- {\sum\limits_{i = 1}^{4}{u_{i}\tau_{i}t}}}} )}$

Currents through R_(SX) _(A) and R_(SX) _(B) illustrate the multipletime constant when superimposed on the same graph. The switching eventsare illustrated along with the switch currents in FIG. 43, whichillustrates the currents through the switch branches in the example RLcircuit of FIG. 42 for 90 degrees of outphasing.

τ₁, τ₂, τ₃, τ₄ are visible for this 90 degree outphase example. Thecurrent through the inductor is the sum of both switch branches and isillustrated in FIG. 44.

At the 180 degrees of outphasing, the power source is continuallyshunted through the switches to ground. This condition eliminates allharmonic currents in the inductor. Only DC currents remain, so that thefundamental is eliminated. This is illustrated in FIGS. 45 and 46. FIG.45 illustrates the current through the switch branches in the example RLcircuit of FIG. 42 for 180 degrees of outphasing. FIG. 46 illustratesthe current through the inductor in the example RL circuit of FIG. 42for 180 degrees of outphasing.

Whenever the outphasing angle<180° the current through the pull upinductor is a ramp:

$i_{A} = {\frac{1}{L}{\int{V_{L}{\mathbb{d}t}}}}$

Therefore, when the switches are closed a voltage is applied across theinductor and a current ramp is generated. The outphasing angle has aneffect of extending the duty cycle of the applied voltage from T_(C)/2up to a full duty cycle, T_(C). As illustrated above at the full dutycycle, T_(C), the switches are completely outphased and the rampdisappears.

Three observations are gleaned from this discussion thus far: a) Thecurrent flowing through the energy storage element (inductor) isapproximately a linear ramp and extends in duty cycle from T_(C)/2 toT_(C)−; b) The current ramp will grow to twice the magnitude as the dutycycle increase, in a linear manner; and c) The average current flowingin the inductor and the resistors increases as the duty cycle of theramp increases up to the 180° outphasing angle, at which time only DCcurrents flow and are equal to a value of the battery voltage divided bythe switch resistance.

a), b), and c) along with the prior differential equation analysis canbe used to examine “purely outphased” MISO, particularly when efficiencyversus power out is examined.

The circuit analyzed above does not possess a real load. Nevertheless, apseudo-efficiency can be defined as follows:

$n_{eff} \approx \frac{{Reactive}\mspace{14mu}{Power}\mspace{14mu}{in}\mspace{14mu}{the}\mspace{14mu}{Fundamental}\mspace{14mu}{Harmonic}}{{Total}\mspace{14mu} D\; C\mspace{14mu}{Power}\mspace{14mu}{Supplied}\mspace{14mu}{to}\mspace{14mu}{the}\mspace{14mu}{Circuit}}$

The Fourier coefficients for the general current ramp with variable dutycycle can be written as:

$a_{n} = {\sum\limits_{n = 1}^{\infty}\lbrack {{\frac{A}{( {n\;\pi} )^{2}}( {{\cos( {( {\frac{1}{2} + \frac{T_{M}}{T_{C}}} )2n\;\pi} )} - 1} )} + {\frac{2\; A}{n\;\pi}{\sin( {( {\frac{1}{2} + \frac{T_{M}}{T_{C}}} )2n\;\pi} )}}} \rbrack}$$\mspace{20mu}{b_{n} = {\sum\limits_{n = 1}^{\infty}\lbrack {\frac{A}{( {n\;\pi} )^{2}} + {\frac{2( {\frac{1}{2} + \frac{T_{M}}{T_{C}}} )A}{n\;\pi}( {1 - {\cos( {( {\frac{1}{2} + \frac{T_{M}}{T_{C}}} )2n\;\pi} )}} )}} \rbrack}}$

where

T_(M) Δ Time Delay associated with outphasing modulation

${\phi\;\underset{\_}{\Delta}\mspace{14mu}{Outphasing}\mspace{14mu}{Angle}} = {{( \frac{2T_{M}}{T_{C}} ) \cdot 180}\mspace{14mu}{degrees}}$

Reactive power is calculated in the fundamental (n=1) and is found toincrease as the duty cycle increased for the ramp. This can be explainedby the ramp amplitude almost doubling, near 180 degrees outphasing.Although the harmonic energy due to the duty cycle decreases slightly,the increase in ramp amplitude over the cycle duration overcomes theFourier losses. FIG. 47 illustrates the increase in the ramp amplitudeof the inductor current for two outphasing angles in the example circuitof FIG. 42.

The reactive power cycled in the inductor is plotted in FIG. 48 versus φwith power in dB_(r) using φ=0° as the reference of zero.

In addition, the pseudo efficiency is given as a function of outphasingangle relative to the efficiency at φ=0°. Notice that pseudo efficiencyis flat up to more than 100° of outphasing. This is further discussedbelow.

For a range of 0 degrees outphasing to nearly 180 degrees of outphasing,the reactive power in the fundamental cycled through the inductoractually increases. In other words, pure outphasing with a MISO node,and nearly ideal switching components, does not produce an attenuationof the available output energy over virtually the entire outphasingangle range. Furthermore, as the current ramp increases in duty cyclefrom 50% to nearly 100%, the DC average increases by 6 dB until justprior to the 100% duty cycle. Again, this is extracted from the Fourieranalysis of the ramped current, revealed by the differential equation.

Although a load circuit is not included for this basic analysis, energyavailable in the inductor can be transferred to a load.

(b) Blended Outphasing in the R-L Case without Load

In the prior section, it was verified by mathematical analysis that thereactive energy transferred to the inductor from the battery at thefundamental frequency does not fall off rapidly as a function ofoutphasing angle. In classical outphasing systems, the power availablewould normally follow a characteristic of:P _(OUT) ∝A ² cos²(φ/2)

In the MISO case, pure outphasing does not produce this result. Rather,the energy available in the inductor increases as was illustrated by thedifferential equation for the simple MISO example.

In this section, the simple example is revisited with the goal ofillustrating how source impedances (more specifically resistances forthis simple example) modify the energy transfer characteristic.

The appropriate differential equation is derived from example circuitshown in FIG. 49.

The relevant circuit equations are:

$\mspace{20mu}{{V_{S} - {L\frac{\mathbb{d}i_{a}}{\mathbb{d}t}} + {( {i_{b} - i_{a}} )\frac{R_{a}}{\sum\limits_{k = 0}^{\infty}{u\lbrack \underset{\underset{u_{a}}{︸}}{( {t - {kT}_{C}} ) - {u( {t - {( {k + \frac{1}{2}} )T_{C}}} )}} \rbrack}}}} = 0}$${{( {{- i_{b}} + i_{a}} )\frac{R_{a}}{\sum\limits_{k = 0}^{\infty}\lbrack {{u( {t - {kT}_{zc}} )} - {u( {t - {( {k + \frac{1}{2}} )T_{C}}} )}} \rbrack}} - {i_{b}\frac{R_{b}}{\sum\limits_{k = 0}^{\infty}\lbrack \underset{\underset{u_{b}}{︸}}{u( {t - ( {{kT}_{C} + T_{M}} ) - {u( {t - {( {k + \frac{1}{2}} )T} + T_{M}} )}} )} \rbrack}}} = 0$

Two system equations result:

${( {\frac{\mathbb{d}}{\mathbb{d}t} - \frac{1}{L( {\frac{u_{b}}{R_{b}} + \frac{u_{a}}{R_{a}}} )}} )i_{b}} = {\frac{1\; u_{b}}{{LR}_{b}}V_{S}}$$i_{a} = {( {1 + \frac{R_{b}u_{a}}{R_{a}u_{b}}} )i_{b}}$

Hence, the general solution can be written as:

${i_{b} = {k_{b}( {1 - {\mathbb{e}}^{{- \underset{\underset{\lambda}{︸}}{(\frac{\frac{R_{a}}{U_{a}} + \frac{R_{b}}{U_{b}}}{L{({1 + \frac{R_{b}R_{a}}{U_{b}U_{a}}})}})}}t}} )}},{t \geq 0}$${i_{a} = {( {1 + \frac{R_{b}u_{a}}{R_{a}u_{b}}} )( {k_{b}( {1 - {\mathbb{e}}^{{- \lambda}\; t}} )} )}},{t \geq 0}$

Initial and final conditions are:

${\begin{matrix}{{@t} = 0} & {i_{b} = 0} \\{{@t} = {0 +}} & {\frac{\mathbb{d}i}{\mathbb{d}t} = \frac{V_{S}}{L}} \\{{@t} = \infty} & {i_{b} = 0}\end{matrix}\therefore k_{b}} = {{\frac{V_{S}}{( {\frac{R_{a}}{u_{a}} + \frac{R_{b}}{u_{b}}} )}\mspace{14mu}{for}\mspace{14mu} R_{a}} = R_{b}}$${{and}\mspace{14mu}{finally}},{i_{a} = {( {1 + \frac{u_{a}}{u_{b}}} )( \frac{V_{S}}{( {\frac{R_{a}}{u_{a}} + \frac{R_{b}}{u_{b}}} )} )( {1 - {\mathbb{e}}^{{- \lambda}\; t}} )}}$

This result is similar to the result presented in the previous sectionyet with a slightly different form. Switch resistances are given by:

$R_{{SX}_{a}} = \frac{R_{a}}{u_{a}}$$R_{{SX}_{b}} = \frac{R_{b}}{u_{b}}$

These resistances are time variant functions. They are also functions ofT_(M).

It was shown in the previous section that unrestrained current in theswitches produces an undesirable outphasing result if the goal is tomimic a classical outphasing transfer characteristic. The availablereactive power in the first harmonic cycled through the inductor can beobtained from:P _(X) _(L) =(0.707i _(a) ₁ )² x _(L)

x_(L) Δ Inductor Reactance

The first harmonic of the current was calculated previously from theFourier analysis in the previous section and is given as:

$i_{a_{1}} = {A( {{a_{1}{\cos( \frac{2\;\pi\; t}{T_{C}} )}} + {b_{1}{\sin( \frac{2\;\pi\; t}{T_{C}} )}}} )}$

a₁, b₁ are the Fourier coefficients and A is an amplitude functionassociated with the gain of the differential equation, given above.

P_(X) _(L) can therefore be tailored to produce an appropriate transfercharacteristic as a function of T_(M) by recognizing that i_(a), is afunction of T_(M). Therefore, the following constraining equation isinvoked:

${P_{X_{L}}( T_{M} )} = {{K\;{\cos^{2}( {\frac{2\; T_{M}}{T_{C}} \cdot \pi} )}} = {( {{.707} \cdot {i_{a_{1}}( T_{M} )}} )^{2}x_{L}}}$

This equation constrains the energy per unit time cycled through theinductor to vary as a function of T_(M) (outphasing delay time)according to the “ideal” classical outphasing transfer characteristic.The equation may be solved in terms of V_(S)(T_(M)), R_(SX) _(a)(T_(M)), R_(SX) _(b) (T_(M)), etc. That is, if the power source and/orthe resistances are permitted a degree of freedom to vary as a functionof T_(M), then the constraining equation possesses at least onesolution.

These equations can be reduced considerably by rewriting and rearrangingterms, using the local time span ramp approximation and associatedFourier analysis:

${A^{2}\{ {{R_{{SX}_{a}}( T_{M} )},{R_{{SX}_{b}}( T_{M} )}} \}} = \frac{K\;{\cos^{2}( {\frac{2\; T_{M}}{T_{C}} \cdot \pi} )}}{( {{{a_{1}( T_{M} )}\cos\frac{2\;\pi\; t}{T_{C}}} + {{b_{1}( T_{M} )}\sin\frac{2\;\pi\; t}{T_{C}}}} )^{2}}$

The left hand side is the portion which must be solved and is related tothe Differential Equation Characteristic.

According to embodiments of the present invention, a numerical solutiontechnique can be applied to obtain R_(SX) _(a) (T_(M)), R_(SX) _(b)(T_(M)). The denominator is a function of the energy of the rampedcurrent pulses cycled in the inductor. a₁ and b₁ are the first harmonicterms of the Fourier component of these pulses. If the properties of thecurrent pulse slope and amplitude can be controlled, then the availableenergy per unit time stored by the inductor can be tailored as well.

Examination of the differential equation for i_(a)(t) gives thefollowing slope calculation:

${{{{{{{{\frac{\mathbb{d}}{\mathbb{d}t}i_{a}}}_{t = 0} = {{{E(t)} \cdot \frac{\mathbb{d}}{\mathbb{d}t}}( {1 - {\mathbb{e}}^{{- \lambda}\; t}} )}}}_{t = 0} + {{( {1 - {\mathbb{e}}^{{- \lambda}\; t}} ) \cdot \frac{\mathbb{d}}{\mathbb{d}t}}{E(t)}}}}_{t = 0}\therefore{\frac{\mathbb{d}}{\mathbb{d}t}i_{a}}}}_{t = 0} = { {( {1 + \frac{u_{a}R_{b}}{u_{b}R_{b}}} )\frac{V_{S}}{( {\frac{R_{a}}{u_{a}} + \frac{R_{b}}{u_{b}}} )}\lambda\;{\mathbb{e}}^{{- \lambda}\; t}} \middle| {}_{t = 0}{{+ \frac{\mathbb{d}}{\mathbb{d}t}}( {( {1 + \frac{u_{a}R_{b}}{u_{b}R_{b}}} )( \frac{V_{S}}{\frac{R_{a}}{u_{a}} + \frac{R_{b}}{u_{b}}} )( {1 - {\mathbb{e}}^{{- \lambda}\; t}} )} )} \middle| {}_{t = 0}{\frac{\mathbb{d}}{\mathbb{d}t}{i_{a}(t)}}  = {{\frac{V_{S}}{L}@t} = 0}}$

Hence, the slope is easy to calculate for the short duration of a singlecarrier cycle time. However, though the slope at t=0 is always constant,the slope at T_(C)/2<t<T_(C) can be modified by significant changes inthe time constant τ_(eff)=R_(eff)/L, where R_(eff) is the effectiveresistance during the cycle. R_(eff) can change as a function of thedesired modulation angle. Unfortunately, the calculation of the Fouriercoefficients for the exponential pulse, though available in closed form,are very tedious expressions to write down and therefore will not bedescribed. The coefficients a₁(T_(M)) and b₁(T_(M)) can be numericallyevaluated.

FIGS. 50 and 51 illustrate how the pulse shape of the current changes asthe switch resistance is permitted to vary significantly for increasingvalues of T_(M). In particular, FIG. 50 illustrates the case of fixedresistance in the switches of 0.1Ω. FIG. 51 provides a control whichsteps the resistance in 10Ω increments from 0.1Ω to 50.1Ω. As can benoted from FIGS. 50 and 51, energy cycling through the inductordecreases according to a true outphasing system when resistance isvaried. The switch resistance or impedance increases with outphasingangle. Both duty cycle and current pulse amplitude vary. This blendedcontrol can be precisely tailored to obtain the ideal outphasingcharacteristic and in fact overcome any other non-linear or parasiticphenomena.

(c) Equations Related to Outphasing with a Load

The previous section dealt with available energy in an inductor cycledby outphased switches possessing internal real valued time varyingimpedances. This section examines a load circuit which is coupled by acapacitor. An example of such circuit is illustrated in FIG. 52.

The detailed transient solution is based on the solution of thefollowing differential equation:

${{\frac{\mathbb{d}^{2}}{\mathbb{d}t^{2}}i\;\ell} + \frac{\gamma_{2}}{\gamma_{1}} - {\frac{\mathbb{d}}{\mathbb{d}t}i\;\ell} - {\frac{\gamma}{\gamma_{1}}i\;\ell}} = {\frac{\mathbb{d}}{\mathbb{d}t}V_{S}}$$\gamma_{0} = ( {{\frac{R_{{SX}_{a}}}{R_{{SX}_{b}}}( {R_{{SX}_{b}} + R_{\ell}} )} + R_{\ell}} )$$\gamma_{1} = {\frac{L}{R_{{SX}_{a}}}\gamma_{0}}$$\gamma_{2} = {{\frac{L}{R_{{SX}_{a}}}( {\frac{1}{C}( {\frac{R_{{SX}_{a}}}{R_{{SX}_{b}}} + 1} )} )} - {\frac{R_{{SX}_{a}}}{R_{{SX}_{b}}}( {R_{{SX}_{b}} + R_{\ell}} )} + \gamma_{0}}$$\gamma_{3} = {\frac{1}{C}( \frac{R_{{SX}_{a}}}{R_{{SX}_{b}}} )}$$R_{{SX}_{a}} = \frac{R_{a}}{u_{a}}$$R_{{SX}_{b}} = \frac{R_{b}}{u_{b}}$

u_(a) and u_(b) were defined previously as the cyclic unit stepfunctions, which are also functions of the outphasing angle (as well astime).

Whenever R_(a) and R_(b) are constant, the differential equation can besolved in a piecewise manner for specific domains of time. The roots arecalculated from:

$\beta_{1},{\beta_{2} = {\frac{- \gamma_{2}}{2\;\gamma_{1}} \pm \sqrt{{\frac{1}{4}( \frac{\gamma_{2}}{\gamma_{1}} )^{2}} - \frac{\gamma_{3}}{\gamma_{1}}}}}$where R_(a), R_(b) are constant.

The current through the inductor can be written as:

$i_{a} = {{\frac{\gamma_{0}}{R_{{SX}_{a}}}i\;\ell} + {\frac{\mathbb{d}}{\mathbb{d}t}( {\frac{1}{C}( {\frac{R_{{SX}_{a}}}{R_{{SX}_{b}}} + 1} )} )i\;\ell}}$

The classical solution occurs whenever both switches are open, whichyields:

$i_{\ell} = {{\frac{V_{S}}{L( {\alpha_{1} - \alpha_{2}} )}( {{\mathbb{e}}^{{- \alpha_{2}}t}{\mathbb{e}}^{{- \alpha_{1}}t}} )} + {{initial}\mspace{14mu}{conditions}}}$$\alpha_{1},{\alpha_{2} = {\frac{- R_{\ell}}{2\; L} \pm \sqrt{{( \frac{R}{L} )^{2}\frac{1}{4}} - \frac{1}{LC}}}}$

Whenever the outphasing angles are small and both switches possess lowvalues for R_(a), R_(b) then the previous analysis for cycling energythrough the inductor applies for the times when switches are closed.When the switches open, the energy from the inductor is released and theload current becomes dominant as indicated by the classical solution.

The analysis is interesting when γ₀, γ₁, γ₂, γ are time variant. It isbest to treat them as quasi-constant over a single charge and dischargecycle. This is reasonable if the values of R_(a) and R_(b) varyaccording to T_(M), the outphasing angle time delay. Fortunately, theenvelope modulation rate is much less than the carrier rate for allpractical scenarios of interest. With this, the first harmonics of thecurrents, i_(l) and i_(a) can be constrained such that;

$\frac{i_{\lambda_{1}}^{2}( T_{M} )}{2\; R_{\ell}} = {A^{2}{\cos^{2}( \frac{\phi}{2} )}}$i_(l) _(i) ² Δ peak of the 1st harmonic (fundamental) flowing throughR_(l)

This is the identical constraint placed on the inductor current given inthe previous section. In that example, the load branch was non existent.The collapsing magnetic field of the inductor transfers energy in thisexample via the capacitor to R_(l). Furthermore, the RLC circuit sets upa resonator with Q. The Q of the circuit in the R-L circuit example islowered to attain the transfer characteristic. As the resistancesR_(a)(T_(M)), R_(b)(T_(M)) are increased to lower the Q on the chargingcycle the Fourier coefficient a₁, b₁ are modified in i_(a) and i_(l) tolower the power transferred to the load.

FIG. 53 illustrates the currents through the inductor and the load inthe example circuit of FIG. 52. FIG. 53 illustrates the cycles involvedfor storing energy in the inductor and the discharge into the load. Notethat the current flowing in the load is filtered considerably by thetuned circuit.

When the switch source resistances vary, outphasing produces the desiredattenuation effect for large angles of φ as illustrated by FIG. 54. FIG.54 illustrates the effect of varying the switch source resistances onthe currents through the inductor and the load in the example circuit ofFIG. 52.

(i) Blended Outphasing Efficiency

This section provides a heuristic explanation for the efficiencyperformance of blended outphasing, as presented in sections 7 and 8.

Using blended outphasing, in every case where modified switchingimpedances are utilized, the minimum switch resistance is selected forthe operating points requiring maximum efficiency. Furthermore, maximumefficiency is tailored according to the waveform envelope statistic.Typically, the probability of low power levels on the signal envelope islow. Therefore, the low efficiency circumstances for the circuit areminimal.

FIG. 56 illustrates a histogram associated with the WCDMA waveform. Asshown, the greatest average probabilities are associated with slightback off from the peaks. At these high levels, the switch resistancesare low and therefore possess relatively small I-R losses. The inductivetank has very high Q's for these low outphasing angles. Occasionally theenvelopes make large excursions or even experience zero crossings. Whileefficiency drops as outphasing increases, these events occurinfrequently.

Efficiency is calculated from the power in the 1^(st) harmonicdissipated by the load divided by the total battery power required tosupply the circuit. The previous section developed the expressions fori_(l) and i₀ which permit the calculation. However, as pointed out inearlier analysis this calculation is not practical in closed form.Therefore, numerical analysis is typically preferred. Thus, efficiencyis usually optimized numerically according to:

$\eta = \frac{( \frac{i_{\ell_{1}}^{2}}{2\; R_{\ell}} )}{( {{\overset{\sim}{i}}_{\ell_{a}} \cdot V_{S}} )}$

FIG. 57 illustrates the power output to the load as T_(M) varies in theexample circuit of FIG. 52.

FIG. 58 illustrates the average DC current from battery in the examplecircuit of FIG. 52

11. Summary

According to embodiments of the present invention, MISOamplifier/combiner circuits driven by VPA control algorithms outperformconventional outphasing amplifiers, including cascades of separatebranch amplifiers and conventional power combiner technologies.Moreover, these circuits can be operated at enhanced efficiencies overthe entire output power dynamic range by blending the control of thepower source, source impedances, bias levels, outphasing, and branchamplitudes. These blending constituents are combined to provide anoptimized transfer characteristic function, where optimization includesseveral aspects, including a well-behaved power transfer characteristic,overall efficiency on a per waveform basis, waveform specificationperformance such as enhanced EVM, ACPR, and unified architecture for allwaveforms, i.e., universal waveform processor.

According to embodiments of the invention, VPA principles include thefollowing.

-   -   a. Power is transferred from the power source to the load with        the greatest expediency. The number of levels of intermediate        processing between the power source and the load is minimized.    -   b. Consistent with a), power is not transferred through a power        combiner.    -   c. MISO inputs are controlled to direct the power flow from the        power source directly to the load, while modifying the spectral        content of the energy as it flows through the node, from the        power supply branch to the output or load branch.    -   d. Various controls are designed to optimize efficiency while        maintaining performance specifications of signals by operating        on the components of the complex envelope.    -   e. The physical structure of the MISO is significantly different        than traditional combiners in terms of impedance, frequency        response, and time delay. These properties permit the effective        application of the principles a) through d). Traditional        combiners cannot easily accommodate principles a), b) or c).

Simulation, mathematical analysis, and lab data agree and indicate thatthe MISO), when combined with the blended control algorithms,outperforms today's complex technologies and approaches efficiencies fargreater than any available technology, while accommodating the everincreasing complex waveforms demanded by the cell phone market.

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. It will be apparent to persons skilledin the relevant art that various changes in form and detail can be madetherein without departing from the spirit and scope of the invention.Thus, the breadth and scope of the present invention should not belimited by any of the above-described exemplary embodiments, but shouldbe defined only in accordance with the following claims and theirequivalents.

What is claimed is:
 1. A method comprising: receiving a plurality ofcontrol signals; generating a plurality of substantially constantenvelope signals from the plurality of control signals and a referencesignal; combining the plurality of substantially constant envelopesignals at a multiple input single output (MISO) node to generate adesired waveform; and controlling the desired waveform at the MISO nodebased on a signal constellation corresponding to the plurality ofcontrol signals.
 2. The method of claim 1, wherein the receivingcomprises receiving the plurality of control signals with an a prioriMISO response for a complex plane.
 3. The method of claim 2, wherein thereceiving the plurality of control signals with the a priori MISOresponse comprises compensating for non-linearities superimposed on thecomplex plane.
 4. The method of claim 1, wherein the generatingcomprises generating the plurality of substantially constant envelopesignals based on a complex envelope of the desired waveform.
 5. Themethod of claim 1, wherein the combining comprises generating a radiofrequency (RF) carrier signal at the MISO node.
 6. The method of claim5, wherein the controlling comprises generating the RF carrier signalbased on a carrier frequency, a modulation angle, an amplitudemodulation, or a combination thereof.
 7. The method of claim 1, whereinthe combining comprises generating the desired waveform based on anEDGE, GSM, CDMA2000, WCDMA, OFDM, or WLAN communication protocol.
 8. Themethod of claim 1, wherein each of the plurality of substantiallyconstant envelope signals corresponds to a respective MISO branch, andwherein the combining comprises combining the plurality of substantiallyconstant envelope signals that traverse their respective MISO branchesat the MISO node.
 9. The method of claim 1, wherein the controllingcomprises controlling an outphasing angle between the plurality ofsubstantially constant envelope signals, a bias at the MISO node, anamplitude at the MISO node, a power supply of one or more MISO devices,an impedance of the MISO node, or a combination thereof.
 10. Anapparatus comprising: one or more modulators configured to receive aplurality of control signals and to generate a plurality ofsubstantially constant envelope signals from the plurality of controlsignals and a reference signal; a multiple input single output (MISO)device configured to combine the plurality of substantially constantenvelope signals at a MISO node to generate a desired waveform; and acontrol circuit configured to control the desired waveform at the MISOnode based on a signal constellation corresponding to the plurality ofcontrol signals.
 11. The apparatus of claim 10, wherein the one or moremodulators are configured to receive the plurality of signals with an apriori MISO response for a complex plane.
 12. The apparatus of claim 11,wherein the one or more modulators are configured to compensate fornon-linearities superimposed on the complex plane.
 13. The apparatus ofclaim 10, wherein the one or more modulators are configured to generatethe plurality of substantially constant envelope signals based on acomplex envelope of the desired waveform.
 14. The apparatus of claim 10,wherein the MISO device is configured to generate a radio frequency (RF)carrier signal at the MISO node.
 15. The apparatus of claim 14, whereinthe control circuit is configured to generate the RF carrier signalbased on a carrier frequency, a modulation angle, an amplitudemodulation, or a combination thereof.
 16. The apparatus of claim 10,wherein the MISO device is configured to generate the desired waveformbased on an EDGE, GSM, CDMA2000, WCDMA, OFDM, or WLAN communicationprotocol.
 17. The apparatus of claim 10, wherein the MISO devicecomprises a plurality of branches, and wherein each branch in theplurality of branches corresponds to each substantially constantenvelope signal in the plurality of substantially constant envelopesignals.
 18. The apparatus of claim 10, wherein the control circuit isconfigured to control an outphasing angle between the plurality ofsubstantially constant envelope signals, a bias at the MISO node, anamplitude at the MISO node, a power supply of the MISO device, animpedance of the MISO node, or a combination thereof.